Moser's worm problem (also known as mother worm's blanket problem) is an unsolved problem in

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

formulated by the Austrian-Canadian mathematician

Leo Moser Leo Moser (11 April 1921, Vienna – 9 February 1970, Edmonton) was an Austrian-Canadian mathematician, best known for his polygon notation. A native of Vienna, Leo Moser immigrated with his parents to Canada at the age of three. He received his ...

in 1966. The problem asks for the region of smallest

area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...

that can accommodate every

plane curve In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic pla ...

of length 1. Here "accommodate" means that the curve may be rotated and translated to fit inside the region. In some variations of the problem, the region is restricted to be

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytope ...

Examples

For example, a circular disk of radius 1/2 can accommodate any plane curve of length 1 by placing the midpoint of the curve at the center of the disk. Another possible solution has the shape of a

rhombus In plane Euclidean geometry, a rhombus (plural rhombi or rhombuses) is a quadrilateral whose four sides all have the same length. Another name is equilateral quadrilateral, since equilateral means that all of its sides are equal in length. The ...

with vertex angles of 60 and 120 degrees (/3 and 2/3

radians The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that c ...

) and with a long diagonal of unit length. However, these are not optimal solutions; other shapes are known that solve the problem with smaller areas.

Solution properties

It is not completely trivial that a solution exists – an alternative possibility would be that there is some minimal area that can be approached but not actually attained. However, in the convex case, the existence of a solution follows from the

Blaschke selection theorem The Blaschke selection theorem is a result in topology and convex geometry about sequences of convex sets. Specifically, given a sequence \ of convex sets contained in a bounded set, the theorem guarantees the existence of a subsequence \ and a co ...

. attribute this observation to an unpublished manuscript of Laidacker and Poole, dated 1986. It is also not trivial to determine whether a given shape forms a solution. conjectured that a shape accommodates every unit-length curve if and only if it accommodates every unit-length polygonal chain with three segments, a more easily tested condition, but showed that no finite bound on the number of segments in a polychain would suffice for this test.

Known bounds

The problem remains open, but over a sequence of papers researchers have tightened the gap between the known lower and upper bounds. In particular, constructed a (nonconvex) universal cover and showed that the minimum shape has area at most 0.260437; and gave weaker upper bounds. In the convex case, improved an upper bound to 0.270911861. used a min-max strategy for area of a convex set containing a segment, a triangle and a rectangle to show a lower bound of 0.232239 for a convex cover. In the 1970s, John Wetzel conjectured that a 30 degree circular sector of unit radius is a cover with area

\pi/12 \approx 0.2618

. Two proofs of the conjecture were independently claimed by and by . If confirmed, this will reduce the upper bound for the convex cover by about 3%.

Notes

References

*. *. *. *. *. *. *. *{{citation , last1 = Movshovich , first1 = Yevgenya , last2 = Wetzel , first2 = John , title = Drapeable unit arcs fit in the unit 30° sector , journal = Advances in Geometry , volume = 17 , year = 2017 , issue = 4 , pages = 497–506 , doi = 10.1515/advgeom-2017-0011 , s2cid = 125746596 , url = https://www.degruyter.com/view/journals/advg/17/4/article-p497.xml , url-access = subscription . Discrete geometry Unsolved problems in geometry Recreational mathematics

Examples

Solution properties

Known bounds

See also

Notes

References